Find the Equation Using Point-Slope Formula (4,3) , (7,7)

$(4, 3)$ , $(7, 7)$

Step 1

Find the slope of the line between $(4, 3)$ and $(7, 7)$ using $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ , which is the change of $y$ over the change of $x$ .

Slope is equal to the change in $y$ over the change in $x$ , or rise over run.

$m = \frac{change in y}{change in x}$

The change in $x$ is equal to the difference in x-coordinates (also called run), and the change in $y$ is equal to the difference in y-coordinates (also called rise).

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Substitute in the values of $x$ and $y$ into the equation to find the slope.

$m = \frac{7 - (3)}{7 - (4)}$

Simplify.

Simplify the numerator.

Multiply $- 1$ by $3$ .

$m = \frac{7 - 3}{7 - (4)}$

Subtract $3$ from $7$ .

$m = \frac{4}{7 - (4)}$

Simplify the denominator.

Multiply $- 1$ by $4$ .

$m = \frac{4}{7 - 4}$

Subtract $4$ from $7$ .

$m = \frac{4}{3}$

Step 2

Use the slope $\frac{4}{3}$ and a given point $(4, 3)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (3) = \frac{4}{3} \cdot (x - (4))$

Step 3

Simplify the equation and keep it in point-slope form.

$y - 3 = \frac{4}{3} \cdot (x - 4)$

Step 4

Solve for $y$ .

Simplify $\frac{4}{3} \cdot (x - 4)$ .

Rewrite.

$y - 3 = 0 + 0 + \frac{4}{3} \cdot (x - 4)$

Simplify by adding zeros.

$y - 3 = \frac{4}{3} \cdot (x - 4)$

Apply the distributive property.

$y - 3 = \frac{4}{3} x + \frac{4}{3} \cdot - 4$

Combine $\frac{4}{3}$ and $x$ .

$y - 3 = \frac{4 x}{3} + \frac{4}{3} \cdot - 4$

Multiply $\frac{4}{3} \cdot - 4$ .

Combine $\frac{4}{3}$ and $- 4$ .

$y - 3 = \frac{4 x}{3} + \frac{4 \cdot - 4}{3}$

Multiply $4$ by $- 4$ .

$y - 3 = \frac{4 x}{3} + \frac{- 16}{3}$

Move the negative in front of the fraction.

$y - 3 = \frac{4 x}{3} - \frac{16}{3}$

Move all terms not containing $y$ to the right side of the equation.

Add $3$ to both sides of the equation.

$y = \frac{4 x}{3} - \frac{16}{3} + 3$

To write $3$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = \frac{4 x}{3} - \frac{16}{3} + 3 \cdot \frac{3}{3}$

Combine $3$ and $\frac{3}{3}$ .

$y = \frac{4 x}{3} - \frac{16}{3} + \frac{3 \cdot 3}{3}$

Combine the numerators over the common denominator.

$y = \frac{4 x}{3} + \frac{- 16 + 3 \cdot 3}{3}$

Simplify the numerator.

Multiply $3$ by $3$ .

$y = \frac{4 x}{3} + \frac{- 16 + 9}{3}$

Add $- 16$ and $9$ .

$y = \frac{4 x}{3} + \frac{- 7}{3}$

Move the negative in front of the fraction.

$y = \frac{4 x}{3} - \frac{7}{3}$

Step 5

Reorder terms.

$y = \frac{4}{3} x - \frac{7}{3}$

Step 6

List the equation in different forms.

Slope-intercept form:

$y = \frac{4}{3} x - \frac{7}{3}$

Point-slope form:

$y - 3 = \frac{4}{3} \cdot (x - 4)$

Step 7

Find the Equation Using Point-Slope Formula (4,3) , (7,7)

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